Estimation and Inference for Minimizer and Minimum of Convex Functions: Optimality, Adaptivity, and Uncertainty Principles
Tony Cai, Ran Chen, and Yuancheng Zhu
-
Abstract:
Optimal estimation and inference for both the minimizer and minimum of a convex regression function under the white noise and nonparametric regression models are studied in a non-asymptotic local minimax framework, where the performance of a procedure is evaluated at individual functions. Fully adaptive and computationally efficient algorithms are proposed and sharp minimax lower bounds are given for both the estimation accuracy and expected length of confidence intervals for the minimizer and minimum.
The non-asymptotic local minimax framework brings out new phenomena in simultaneous estimation and inference for the minimizer and minimum. We establish a novel Uncertainty Principle that gives a fundamental limit for any convex regression function to how well the minimizer and minimum can be estimated simultaneously. A similar result holds for the expected length of the confidence intervals for the minimizer and minimum.
- Paper: pdf file.